On 03/24/2011 01:23 PM, Tim Lahey wrote:
On 03-24-2011, at 11:18 AM, SherjilOzair wrote:
Cool ideas, indeed. Very useful, I'm sure, and quite easy to
implement.
For the abstract Matrix part, A new constructor-less data-less class
AbstractMatrix which inherits class Matrix. This class' objects will
serve as pointers to Matrices. And the object would later be made to
point to a concrete class. We can also make the other concrete classes
sub-classes of this class, but that remains to be seen. Another slight
modification will need to be done to the solve function, to solve
simple eqns like Ax=B, where x is the abstract vector.
I'm not sure if this is the best idea. Working with these kind of matrices
symbolically is quite different than regular matrices. They don't have any
components, you're just working with the matrix as a whole ignoring the
internal details. So, one won't have some of the functions of a regular matrix.
For instance
1/2*a.transpose()*N*N.transpose()*a
where a and N are vectors. Then when you differentiate with respect to a, you'd
get,
transpose((1/2*N*N.transpose().a)+(/2*a.transpose()*N*N.transpose()
which then becomes,
1/2*a.transpose()*transpose(N*N.transpose()) + 1/2*a.transpose()*N*N.transpose()
in this case, because of the N*N.transpose() will be symmetric, these can be
added together to get,
a.transpose()*N*N.transpose()
Note that this works even not knowing what the vectors are as long as they are
of the right shape.
The tricky part is working with the parts and taking transpose operations to
rearrange things in the right order. I don't know how possible that is to do
automatically.
I've put a PDF of the Maple worksheet I have that does this at,
http://db.tt/QBF5Eaz
and for those with Maple, the actual worksheet is,
http://db.tt/6jJoFdO
The second idea is simpler, Just make the matrix class accept Matrices
as elements. We'll have to put in a flag though, a boolean
contains_matrices, which will tell particular methods to compute the
results with keeping this in mind, OR
An alternate design : Another class BlockMatrix, which will accept
matrices as elements, but when any computation is needed, it will call
its BlockMatrix.expand() function, whose result will be feeded to the
regular matrices functions. The expansion will be stored in a local
cache in the object, to be re-used.
The block matrix is a special kind of matrix, that has the "abstract" matrices as
components. Basically, you're opening up an "abstract" matrix to provide some structure,
but each of those parts is left as a black box.
Thank you Tim. If you're interested to do the Linear Algebra module,
as Aaron says there is plenty of work to be done. We can discuss more
of it on IRC or mail.
You're welcome. I tend to think of these as kind of parallel to the Linear
Algebra module since these are only working with matrices as a whole or in
large parts. Once as much can be done with these, you'd create the details of
the individual matrices and then substitute for the abstract/block matrices to
get the final result.
Cheers,
Tim.
Instead of abstract matrix how about abstract tensor such as in the
Penrose notation.
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